A numerical approach to the flux density integral for reflected sunlight

Abstract A computer model of the central receiver system must evaluate the flux density on the receiver due to sunlight reflected by the heliostats in the collector field. Several approaches are available but each has its limitations. The Monte-Carlo approach represents all of the heliostat behavior but is relatively slow in terms of CPU time and is not suitable for optimization purposes. FLASH is an analytically exact approach for flat polygonal heliostats but is slow and not applicable to dished heliostats or aureole effects. Cone optics programs evaluate the flux density by a direct numerical integration of the double integral, but this method is very slow if accuracy is required. HCOEF is a two dimensional Hermite polynomial method which is relatively fast and can be extended to include canting, focusing, solar limb, and guidance error effects. However, the polynomial approximation breaks down for near heliostats, small guidance errors, and aureole effects. The new image generators based on KGEN overcome this limitation, but running times compare to FLASH and are 3 or 4 slower than HCOEF. The new approach proposed in this study assumes isotropic gaussian guidance errors. Hence, the flux density integral reduces to several iterated single integrals which can be precalculated and stored in a table for interpolation as needed. The LBL solar telescope data are fed into a convolution integral which represents the guidance errors. Aureole effects can be switched on or off at this point. A vector of convoluted solar data is input to another integration which gives the table of normalized flux contributions. The tabular values depend on the position of the flux point with respect to an edge of the heliostat as seen in the image plane. The image map of the heliostat is linear unless ripples or irregularities occur; hence, effects due to canting and dishing can be included by a ray trace of the heliostat vertices. The use of tabular interpolation is not as fast as expected because of the time required to calculate the distance between the flux point and the image of the vertices. The accuracy of this method is limited by interpolation errors, and better results can be obtained with the same CPU time if more core is used for a larger table. It is possible to eliminate the table by introducing a Romberg type of integrator which bisects the interval until sufficient accuracy is achieved; however, this approach is inefficient unless the images are relatively small compared to the receiver. The convolution process in KGEN is fast and can be used to calculate moments for HCOEF and coefficients for FLASH which utilize the LBL data.